Thinking Steadfast as a Dog

"Ow, damn it, my train of thought is broken!"

Doing math problems is just like writing a novel. When the flow is smooth, you can write ten chapters a day; when the flow is broken, it's one chapter in ten days.

Just as Shen Qi was about to crack the key to the first Geometry problem, the Vice President interrupted his train of thought.

"This uncle is so annoying." Shen Qi had to re-organize his thoughts, which cost him an extra five minutes.

The proof filled an entire sheet of white paper, and Shen Qi finally found the value of sinψ.

The answer surprised him; sinψ turned out to be 1/2, which corresponded to a 30-degree angle.

He measured it with a ruler, and it seemed to be 30 degrees.

He verified it using Lobachevsky's graphing method, and it was indeed 30 degrees.

"I'm stupid, I'm really stupid..." Shen Qi realized a low-level mistake. He had been confused by the complex geometric patterns, and the forward deduction had taken nearly an hour.

If he had directly calculated the degree of ψ using Lobachevsky's graphing method first and then verified backward that this angle ψ was 30 degrees, he could have saved at least half the time.

Of course, Lobachevsky's graphing method couldn't be used on the exam paper; drawing on scratch paper was fine.

Verifying a conclusion with the conclusion in hand was easier than deriving an unknown.

This was a game, a mathematical game between the question setter and the solver.

War is the best training for soldiers; exams are the best review.

Although he had the system's assistance, Shen Qi gained a new understanding of mathematics in this extremely difficult semi-finals.

There must be a deeper relationship between assumption and proof. Truth or fallacy is not as simple as it appears on the surface. Opposition, fallacy, perhaps it's a backward deduction of truth.

Shen Qi's mind wandered. He thought for a long time, several minutes actually. It seemed his math level was still too low; he couldn't figure out many things.

"There are still two hours left for the exam. All contestants, please hurry up and answer the questions." A proctor said to all the contestants.

"Two hours, two problems left." Shen Qi snapped back to reality and moved on to the next problem.

This semi-final exam paper was the one with the fewest problems he had ever done, only three.

It was also the most difficult. The first problem, which had the lowest point value, took Shen Qi an hour.

The second problem was an Algebra question, and the prompt was as follows:

1

1-1

1-2-1

1-3-3-1

1-4-6-4-1

1-5-10-10-5-1

1. Please calculate the sum of all numbers in the 1024th row. (5 points)

2. And prove that the case where any number in the 4201st row is a fraction or a negative number is applicable. (15 points)

In fact, many high school students were familiar with this number triangle. Who didn't know Yang Hui's Triangle? Middle school students who participated in the Math League or Math Olympiads all knew the patterns of Yang Hui's Triangle.

Shen Qi certainly understood this number triangle. This number triangle was called Yang Hui's Triangle in China and "Pascal's Triangle" in the West, named after two mathematicians from China and the West respectively.

The pattern of Yang Hui's Triangle was not hard to observe: each number in the triangle was the sum of the two numbers directly above it.

Following this pattern, Shen Qi quickly calculated the sum of all numbers in the 1024th row as xxxxx... It was an astronomical number, expressed as 2 to the power of 1023.

The first part of the second question was practically a freebie, hence the low score of only 5 points.

The difficulty lay in the second part, worth 15 points.

Proving forward that the case where any number in the 4201st row is a fraction or a negative number is applicable was very troublesome; he had no idea where to start, finding no clues at all.

Shen Qi wanted to work backward. The content required to be proven in the second part must have a formula, theorem, or corollary as its basis.

"Bernoulli's permutations and combinations, or probability theory? No, that doesn't seem right."

"Vieta's expansion formulas for three special types of equations? Not that either."

"When it comes to pure number games like this, Fermat was a top player. That's right, it must be Fermat. He was good friends with Pascal, and they often corresponded. This question is based on Pascal's Triangle."

"Fermat proposed hundreds of conjectures in his lifetime, and 99% of them were proven to be true by later generations. He was called the 'King of Amateur Mathematics,' but I absolutely do not believe Fermat's mathematical level was amateur like mine."

"My head hurts. Of Fermat's 273 conjectures, I've only studied about 70 at most. Which one could it be? Does it touch upon a blind spot in my mathematical knowledge?"

Shen Qi put down his ballpoint pen, closed his eyes to rest, and racked his brain for a solution.

The Vice President ambled over to Shen Qi's back again, a smug smile on his face. He was quite pleased with Shen Qi's current state: "Young man, even if you could solve the first Geometry problem and are familiar with the patterns of Yang Hui's Triangle, so what? The second part of my question, its difficulty is beyond your imagination, isn't it?"

Just then, Shen Qi's eyes suddenly opened, shining brightly: "I've got it! This is Fermat's (1+a) corollary! He said this corollary was definitely true and didn't need to be proven, but 100 years after his death, his compatriot proved it."

The idea is here, the idea is here!

Shen Qi's inspiration gushed out like a spring. He picked up his ballpoint pen, ready to solve the problem.

Just as he was about to put pen to paper, Shen Qi inexplicably felt a chill coming from behind him, exactly like before.

Shen Qi glanced back, extremely displeased: "What the hell, you again! Are you just passing by again?"

The Vice President feigned calmness. "The exam hall is just this big. As an invigilator, I occasionally walk by." He then left with his hands behind his back.

"Uncle, you're really annoying!" Shen Qi calmed himself down as quickly as possible and began working on the proof for the second part of the second question.

Ten minutes later, Shen Qi had written out the complete proof for the second part of the second question. Verified by Fermat's (1+a) corollary, the proof was completely valid, applicable to any number in the 4201st row being a fraction or a negative number.

"Both the first and second major questions have been solved. 35 points secured."

With two questions solved consecutively, Shen Qi's confidence soared. He began working on the final question.

The Vice President was a man who couldn't sit still. After strolling around, he doubled back and secretly observed the other contestants from behind.

"He Tao, this young man isn't doing well. Besides knowing Yang Hui's Triangle, he can't even start the other problems."

"Luo Ming, from Nangang City Foreign Language School. As expected of a student from a prestigious school, this young man is quite capable. He solved the first Geometry question correctly, but his proof for the second part of the second question is clearly wrong. Young man, you're going in the opposite direction, a directional error."

The Vice President estimated that Shen Qi was the contestant with the fastest progress and highest accuracy in the entire exam hall. He couldn't help himself, he really couldn't, so he once again crept up behind Shen Qi to observe secretly.

Shen Qi suddenly turned around, on the verge of a breakdown. "Uncle, can you please stop doing that? Why do you have to make things difficult for me?"

The Vice President looked quite innocent. "I created this exam paper. I'm just taking a look. Is it not okay if I don't say anything?"

"Holy crap, you made this paper?" Shen Qi never expected that the master question-setter in his mind would turn out to be this sleazy uncle.